143.
b. Find the average value of ln(x) over [1, e].

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Recall the formula for the average value of a function \(f(x)\) over the interval \([a, b]\): \[\text{Average value} = \frac{1}{b - a} \int_a^b f(x) \, dx\]

Identify the function and interval: here, \(f(x) = \ln(x)\), \(a = 1\), and \(b = e\).

Set up the integral for the average value: \[\frac{1}{e - 1} \int_1^e \ln(x) \, dx\]

To evaluate the integral \(\int \ln(x) \, dx\), use integration by parts. Let \(u = \ln(x)\) so that \(du = \frac{1}{x} dx\), and let \(dv = dx\) so that \(v = x\).

Apply integration by parts formula: \[\int u \, dv = uv - \int v \, du\] which gives \[\int \ln(x) \, dx = x \ln(x) - \int x \cdot \frac{1}{x} \, dx = x \ln(x) - \int 1 \, dx = x \ln(x) - x + C\] Use this to evaluate the definite integral from 1 to \(e\) and then substitute back into the average value formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Average Value of a Function

The average value of a continuous function f(x) over an interval [a, b] is given by (1/(b - a)) times the integral of f(x) from a to b. It represents the mean height of the function on that interval.

Definite Integral

A definite integral calculates the net area under the curve of a function between two points a and b. It is essential for finding accumulated quantities, such as total area or average values over intervals.

Integration of the Natural Logarithm Function

Integrating ln(x) requires integration by parts, using u = ln(x) and dv = dx. The integral of ln(x) is x ln(x) - x + C, which is crucial for evaluating the definite integral of ln(x) over [1, e].

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